The above picture is more complicated in the modular case (that is,
over fields of positive characteristic), see [1]}, [2]}, [3]}, but
\(\mathfrak {m}_0\) , \(\mathfrak {m}_2\) , \(\mathfrak {V}\)  and their truncations always show up. We refer the
reader to the book [4]} for a general treatment of modular Lie
algebras. In this paper, we show that if the field \(\mathbb {F}\)  has
characteristic \(p\ge 5\) , then the Lie algebra \(\mathfrak {m}_2(p)\)  admits a
family of restricted Lie algebra structures \(\mathfrak {m}_2^{\lambda }(p)\) 
parameterized by elements \(\lambda \in \mathbb {F}^p\) . We describe the
isomorphism classes of these algebras, calculate the ordinary and restricted cohomology spaces with trivial coefficients \(H^q\big (\mathfrak {m}_2^{\lambda }(p)\big )\)  and \(H_*^q\big (\mathfrak {m}_2^{\lambda }(p)\big )\)  for \(q=1,2\)  and give explicit bases for those spaces. We also give the bracket
structures and \([p]\) -operations for the corresponding restricted
one-dimensional central extensions of these restricted Lie algebras.
